The average force required to stop a bullet of mass 16 g moving at a speed of 260 m/s, when it penetrates a block of wood to a distance of 12 cm is:

Question

Accepted Answer

To determine the average force required to stop the bullet, we can use the work-energy principle. The work done by the force to stop the bullet is equal to the change in kinetic energy of the bullet.

1. **Calculate the initial kinetic energy (KE) of the bullet:**

The formula for kinetic energy is:
   $$
   KE = \frac{1}{2}mv^2
   $$
   where $ m $ is the mass of the bullet and $ v $ is its velocity.

Given:
   - Mass $ m = 16 \, \text{g} = 0.016 \, \text{kg} $ (converted to kilograms)
   - Velocity $ v = 260 \, \text{m/s} $

$$
   KE = \frac{1}{2} \times 0.016 \, \text{kg} \times (260 \, \text{m/s})^2 = \frac{1}{2} \times 0.016 \times 67600 = 540.8 \, \text{J}
   $$

2. **Calculate the work done by the force:**

The work done by the force is equal to the change in kinetic energy, which is the initial kinetic energy since the final kinetic energy is zero (the bullet stops).

$$
   \text{Work} = 540.8 \, \text{J}
   $$

3. **Relate work to force and distance:**

The work done by the force is also given by:
   $$
   \text{Work} = F \times d
   $$
   where $ F $ is the force and $ d $ is the distance over which the force acts.

Given:
   - Distance $ d = 12 \, \text{cm} = 0.12 \, \text{m} $ (converted to meters)

Solving for $ F $:
   $$
   F = \frac{\text{Work}}{d} = \frac{540.8 \, \text{J}}{0.12 \, \text{m}} = 4506.67 \, \text{N}
   $$

The negative sign indicates that the force is acting in the opposite direction to the motion of the bullet (decelerating it).

4. **Justification for Option 2:**

The calculated force is approximately $-4510 \, \text{N}$, which matches Option 2. This option correctly represents the average force required to stop the bullet, considering the direction of the force is opposite to the bullet's motion.

5. **Explanation for why other options are incorrect:**

- **Option 1 (-9020 N):** This value is approximately double the calculated force, which is not supported by the work-energy calculations.
   - **Option 3 (9020 N):** This value is incorrect because it not only has the wrong magnitude but also the wrong sign, indicating a force in the same direction as the bullet's motion, which is not possible for stopping the bullet.
   - **Option 4 (4510 N):** Although the magnitude is correct, the absence of a negative sign does not accurately represent the direction of the force, which should be opposite to the bullet's motion.

Thus, Option 2 is the correct answer, as it accurately reflects the magnitude and direction of the force required to stop the bullet.